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So welcome to lecture number four of the quantum mechanics course. Today we talk about projectors,

bras and kets and projectors play a key role in quantum theory and in fact we already saw

that in the axioms and they deserve to be studied in detail and today we'll do that

and essentially this is very cheap linear algebra up to the subtlety that we may deal

with infinite dimensional subspaces and there is a little subtlety we have to take care

of there. So most of the first part of today's lecture will be rather trivial up to one or

two points which I would like to draw your attention to. That's the first topic of the

lecture. The second topic of the lecture are the so called bras and kets. So that's the

Dirac bra and ket notation and their definition is very clear and is based on the so called

Riesz lemma that we're going to prove. Riesz like this on the Riesz lemma. Their use in

quantum mechanics is very intuitive and that's the reason why we call it the Riesz lemma.

Their use in quantum mechanics is very intuitive and that's the reason why people like it because

you can write projectors using the bra ket notation in irresistibly charming simple form.

Well, while their use is intuitive, its definition, the definition of that use is rather cumbersome.

So if you really want to say how to use them, then you have to invent all kinds of extra

rules of how you deal with it and what you really mean by it. And at the end of the day,

all of this is already in our standard notation without inventing lots of standard rules,

of extra rules. So we will not use that notation. So apart from an initial charm it carries,

it only causes trouble and misunderstanding and wrong intuitive conclusions and all of

that. We will not use that notation but obviously I introduce it because the textbooks are

covered with this stuff and of course people informally talk using the bras and kets but

you will always want to translate this into proper notation and then you say, oh is that

what you mean? Oh now I understand. Okay. Or if that's what you mean, well then I don't

understand. Okay. So we have a fully developed notation. We don't need extra notation with

hidden rules. Okay. As intuitive as they may seem. So we first start with projections

and Pythagoras theorem. Now obviously in finite dimensional Hilbert spaces, vector spaces,

there's not much to be said. Nevertheless, we will consider a separable Hilbert space.

So let H be a separable Hilbert space. Should I ever in the future fail to mention separable,

I meant to say it, that H be a separable Hilbert space with the appropriate structure on it.

Then we have the following definition, very simple definition. Let psi in H be, well let

psi be in H, then and E in H a unit vector, meaning of course that the norm is one. Then

we define psi parallel, namely parallel to E, as that part of psi which you obtain by

projection to E in this fashion. So that's just a definition. It's called the projection

of psi to E. It's very simple. And whatever is left, so if you subtract from psi this

projected part, you call the orthogonal complement. Okay, so this is really, this is really very

simple. Now here maybe it's important to note the order the psi comes second so that this

projection map taking psi to psi parallel is linear. If the psi was in the first slot,

it would be anti-linear, right? So please keep the order in mind. Okay, so now this

is very simple. This is for one vector, and it's interesting to understand this for an

entire, well at most countable set of orthogonal vectors E. It's interesting to understand

the extension of this elementary or these elementary definitions to at most countable sets E, I,

where the I comes from an at most countable set capital I. And obviously if we're dealing

with a separable Hilbert space, this orthonormal countable set may well be its basis. So this

definition comes in the form of a proposition because one cannot simply define here, one

needs to prove that all the limits exist. So a proposition, again, let psi be in H and

let the E, I's with I from some I be an orthonormal set in H, so not necessarily a basis yet,

for an at most countable set index set I. Then we have that A, well you can still decompose

psi or you can decompose psi as psi orthogonal plus psi parallel, where the psi parallel

is defined as the sum of over all the I in I of projecting the psi to the Ith element